Show that the relation R defined in the set A of all polygons as R = {(P 1 , P 2 ): P 1 and P 2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question

Philoid · Accepted Answer

It is given that the relation R defined in the set A of all polygons as

R = {(P₁, P₂): P₁ and P₂ have same number of sides},

Then, R is reflexive since (P₁, P₂) ϵ R as the same polygon has the same number of sides with itself.

Let (P₁, P₂) ϵ R

⇒ P₁ and P₂ have the same number of sides.

⇒ P₂ and P₁ have the same number of sides.

⇒ (P₂, P₁) ϵ R

Therefore, R is symmetric.

Now, let (P₁, P₂), (P₂, P₃) ϵ R

⇒ P₁ and P₂ have the same number of sides. Also, P₂ and P₃ have the same number of sides.

⇒ P₁ and P₃ have the same number of sides.

⇒ (P₁, P₃) ϵ R

Therefore, R is transitive.

Thus, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have 3 sides.

Therefore, the set of all elements in A related to triangle T is the set of all triangels.

Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Show that the relation R defined in the set A of all polygons as R = {(P₁, P₂): P₁ and P₂ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?